What I am doing now is simple. I have a random series ##f(r)##, with boundary condition ##f(0)=0## and ##f(1)=0##. I want to approximate it using ##J_1(\alpha_{1n}r)##, which satisfies the B.Cs.
I had a plan to solve a system of ##nr## linear equations numerically, where ##nr## is the number...
In my case, I do not know which order of Bessel functions should be used as the basis functions.
In many other cases, the original profile ##f(r)## contains Bessel functions of different orders. I am wondering how can I decompose it with ##J_m(r)## with fixed ##m =1##?
Orthogonality condition for the 1st-kind Bessel function J_m
$$\int_0^R J_m(\alpha_{mp})J_m(\alpha_{mq})rdr=\delta_{pq}\frac{R^2}{2}J_{m \pm 1}^2(\alpha_{mn}),$$
where α_{mn} is the n^{th} positive root of J_m(r), suggests that an original function f(r) could be decomposed into a series of 1-st...